Torque Estimation Algorithms for Stepper Motor. Master s Thesis in Electrical Engineering JULIA JANSSON CHARLOTTE RÅDAHL

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1 Torque Estimation Algorithms for Stepper Motor Master s Thesis in Electrical Engineering JULIA JANSSON CHARLOTTE RÅDAHL Department of Electrical Engineering CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2018

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3 Master s thesis 2018: EENX30 Torque Estimation Algorithms for Stepper Motor Julia Jansson Charlotte Rådahl Department of Electrical Engineering Chalmers University of Technology Gothenburg, Sweden

4 Torque Estimation Algorithms for Stepper Motor JULIA JANSSON CHARLOTTE RÅDAHL JULIA JANSSON, CHARLOTTE RÅDAHL, Supervisor: Examiner: Stefan Lundberg, Electric Power Engineering Richard Kahl, Sigma Embedded Engineering Stefan Lundberg, Electric Power Engineering Master s Thesis 2018:EENX30 Department of Electrical Engineering Chalmers University of Technology SE Gothenburg Telephone Cover: Valve control system consisting of a butterfly valve, gearbox and 2-phase bipolar hybrid stepper motor. iv Gothenburg, Sweden 2018

5 Torque Estimation Algorithms for Stepper Motor JULIA JANSSON CHARLOTTE RÅDAHL Department of Electrical Engineering Chalmers University of Technology Abstract Preventive health monitoring solutions of valve control systems for offshore applications are profitable due to possibility to plan maintenance work. Load torque is a good indicator of system health as it provides information to the user whether the controlling motor can generate enough torque to move the valve stem or if it is close to its limits. To do this it is desirable to avoid expensive and large torque sensors, opening up for the opportunity to investigate the possibility of estimating the torque on the load based on motor parameters. The motor under investigation is a 2-phase bipolar hybrid stepper motor with torque components from both the variable reluctance and the permanent magnet in the rotor. Measurements confirm neglecting the reluctance torque component to be acceptable, simplifying the derived motor torque expressions. Based on a Simulink model of the system developed from mathematical expressions regarding motor and load together with measured motor specific parameters, two motor torque estimation methods were developed and compared. Simulations of the model were executed to estimate uncertainties in the load torque algorithm. The load torque estimate was in turn depending on motor torque algorithm simplifications in order to limit the number of sensors in a implementation while upholding a approved tolerable accuracy. An estimation of the load torque was presented based on the motor torque calculations derived from output power and the assumption of a constant speed. The error of this estimate during a worst case scenario is smaller than 13% having a total load torque above 2 Nm, a value continuing to decrease with an increased load torque. In an implementation of this algorithm, current sensors are the only necessary sensors while remaining parameters are extracted from the reference values. The current sensors are used in systems having a closed loop current control, making the suggested load torque estimation not requiring any added sensors in an implementation. Keywords: Load Torque Estimation, 2-phase Bipolar Hybrid Stepper Motor, Butterfly Valve, Motor Torque, Simulink Model. v

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7 Acknowledgements First and foremost we want to show our gratitude to Stefan Lundberg, our supervisor at Chalmers University of Technology, for a tremendous patience when guiding us through the snowy forest of motors. Without your knowledge and advise we would not have learned half as much. We would also want to thank Richard Kahl and Sigma Embedded Engineering for advice, feedback and the opportunity to write this thesis. Without your support we would have never had this much fun while learning at the same time. An additional thank to LK Valves for always taking the time to answering our questions and providing this thesis with pictures of the valve. We hope our work will be in your favour in order to prevent valve failures. Julia Jansson and Charlotte Rådahl, Gothenburg, June 2018 vii

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9 Contents 1 Introduction Background Aim Limitations Valve Control System The Stepper Motor Hybrid Stepper Motor Electrical Model Of The Hybrid Stepper Motor Mechanical Model Of The Motor And Load Controller and Drive Circuit The Valve And Gearbox Load Torque Estimation Phase Stepper Motor Torque Expression Motor Torque Estimation Using Power Motor and Measurement Setup Stepper Motor System In Thesis Motor Parameter Measurement Setup Winding Resistance Maximum Flux Linkage Position Dependent Inductance Detent Torque Measurement Setup Extracted Stepper Motor Parameters Motor Torque Measurements Winding Resistance Maximum Flux Linkage Position Dependent Inductance Detent Torque Simulink Model Valve Control System model implementation Motor Block Mechanical System Block Detent Torque Block ix

10 Contents Load Torque Block Simulation Results Verification Of Model Motor Torque Estimation Comparison Of Motor Torque Equations Supply Voltage And Speed Variation Resistance Variation Rotor Speed And Position Sensor Resolution Reference Versus Simulated Rotational Speed Load Torque Assuming Constant Speed Load Torque Estimation During Worst Case Operation Discussion Conclusion 59 Bibliography 61 x

11 1 Introduction 1.1 Background The stepper motor is a device generating a stepwise rotational motion, which has an advantage in applications where a precise position control is of importance [1]. A microcontroller determines the sequence to produce winding currents and consequently a magnetic field in different phases, creating a rotational movement of the motor shaft by defined steps [2]. A result of the microcontroller is the stepper motors ability to be well functioning as an open loop system. In a closed loop system the number of steps achieved by the motor could be counted and made sure to be exact, an option of accuracy however is not necessary in all applications [3]. Due to its capability of creating sequential steps, the stepper motor has found application in for example computerised numerical control machines (CNC) and disk drives [1, 4]. Another load controlled by a stepper motor could be a valve. In applications requiring a lightweight system the butterfly valve is of interest. The flow of fluids in a butterfly valve is regulated through a disk opening and closing through a quarter turn, with a rubber lining ensuring a minimisation of leakages. Every opening and closing will tear the valve and as the number of openings and closings depends on the application the valve lifetime is difficult to estimate. One option to determine the mechanical health of the valve is to measure the torque required to alter the position of the valve. A system with a preventive health monitoring method could prevent failures and limit subsequent material and economical damage. Furthermore, a product informing the user when reparation is required give rise to a new business model. The supplier can sell the product as a subscription instead of an one time cost, opening an opportunity to secure the relationship between the supplier and buyer by signing more extensive contracts. In addition, unnecessary substitution of valves can be prevented due to the increased knowledge of each products individual lifetime. Replacements could be planned and performed at a time most suitable for the company, an interesting solution concerning industries directed towards offshore applications such as oil rigs and ships due to the limited maintenance availability. The torque produced by the motor to close and open the valve can be measured using a torque sensor. However, adding parts to the system also increase the number of damageable components. Together with increased cost and space it is therefore important to limit the number of system components. To minimise these factors, the torque could instead be estimated from the stepper motor. This is a favourable method as it would utilise the existing hardware and the estimator could 1

12 1. Introduction be implemented in the controller program, compared to the bulky torque sensor. 1.2 Aim In this master s thesis a Simulink model of a specific stepper motor and load system is developed with motor parameters extracted through measurements. Based on the model at least two algorithms estimating the load torque without using torque sensors will be constructed. An evaluation of the load torque estimation will be performed, with respective algorithms compared to each other to establish the most suitable choice for the given valve control system. 1.3 Limitations The design of the stepper motor, its controller and the valve will not be altered or optimised with respect to the scope of this thesis. Neither will considerations regarding different materials going through the valve be taken into account, and thereby their effects on the system behaviour. This due to that all system components are not, in the beginning of the project, decided by the company. The system will be modulated to consider a specific stepper motor. However, by changing the motor specific parameters the model can be applied to other stepper motors. The remote diagnosis application of the system where the load torque estimation algorithms are developed in an embedded application, will not be designed in this thesis. The conclusions will be based on results provided by a simulation environment and verification in a real system will not be performed. 2

13 2 Valve Control System This master s thesis regard a valve control system consisting of a butterfly valve controlled by a stepper motor. A schematic of the valve control system is shown in Figure 2.1. The figure shows a power supply that feed a DC voltage to a drive circuit regulated by a control circuit. Depending on the controller signals the windings of the 2-phase stepper motor are energised in a specific order. The energised phases A (A+, A ) and B (B+, B ) are seen in Figure 2.1 as a link between the drive circuit and the 2-phase stepper motor. The current in the windings generate a magnetic field which produce a torque that rotates the rotor. Through the gearbox the mechanical torque is transformed to a preferred rotational force. This up scaled force is used to open or close the valve and in turn control the flow. In the following sections the components of the valve control system are described more in detail. Figure 2.1: System components together with significant signals and their direction. 2.1 The Stepper Motor The stepper motor is defined as a polyphase synchronous motor with both stator and rotor having teeth consisting of a magnetic permeable material, making the stepper motor salient as the source of flux depend on the rotor angular position [3, 5]. Shown in Figure 2.2 is the construction of the rotor, where the teeth, motor shaft and the polarities are defined. With an advantage of low cost and high reliability the stepper motor is additionally well functioning in open loop systems [2, 5]. As 3

14 2. Valve Control System the name implies the stepper motor generates a stepwise rotational movement of the motor shaft. The motor drive is regulated by a controller designed to transmit logic signals to create the most suitable stepwise rotation for the given application [6]. Figure 2.2: Construction of the hybrid stepper motor rotor. The opposite polarity of the permanent magnet rotor can be noted as well as the rotor shaft and the grey rotor teeth. The variable reluctance, permanent magnet and a mixture of the two, called hybrid, are the three most common types of stepper motors. A hybrid stepper motor practice both the source of flux principle from the variable reluctance motor together with the principle of the permanent magnet. Applying the principle from the permanent magnet, the flux components considers both the stator and the permanent magnets in the rotor [2]. With a current through the stator windings the stator teeth are polarised. As a result of their polarisation and the permanent magnets in the rotor, a field between the windings and rotor teeth occur moving the rotor. Consequently the torque depend on the field strength resulting from the flux components, and the angle between the fluxes [2]. Observed in Figure 2.3 is the distribution of the rotor teeth of a hybrid stepper motor. Energising phase B so that B+ becomes a north pole, and B- a south pole, will attract the rotor teeth and lock the position shown in Figure 2.3. Based on that position, energising phase A so that A+ has a north polarity and A- becomes a south pole will create a counterclockwise rotational force, aligning the closest south tooth with the positive A-phase and the closest north tooth with the negative A-phase. A variable reluctance motor instead use the principle of reluctance to generate torque. Ohm s law for magnetic circuits is expressed as mmf = ΨR, (2.1) and describes the relation between the magnetic field force mmf, the magnetic flux Ψ and reluctance R [2]. Reluctance is a magnetic property equivalent to resistance in electrical circuits, i.e. the magnetic flux will flow through the path with least magnetic reluctance [7]. The working principle of the contribution from the reluctance motor can be described by neglecting the magnetic poles in Figure 2.3, i.e. 4

15 2. Valve Control System Figure 2.3: Phase distribution of the rotor of a hybrid stepper motor. The figure shows the rotor edge coinside with the shaft. assuming the rotor not to be magnetised. The rotor is constructed of a permeable material, therefor if phase B is energised, the rotor would be locked to the position shown in the figure as the position provides the smallest air gap between the stator and rotor teeth, resulting in a lower reluctance compared to the reluctance in a larger air gap [2, 8]. If instead a current is applied to phase A the rotor would turn either clockwise or counterclockwise in order to align the rotor teeth with phase A. However, with the simple motor structure provided by Figure 2.3 the rotational direction can not be determined since the distance between the closest rotor teeth and the A phase is equal. A hybrid stepper motor use both the characteristic of the permanent magnet motor together with the variable reluctance stepper motor. It therefor use the aspiration from the permanent magnet rotor teeth to attract or repel the excited windings as well as creating a rotation in order to minimise the air gap between stator and rotor, and thus obtain minimum reluctance [9, 10]. For applications requiring high motion resolution a small step angle of the rotor is desired, as it denotes the smallest individual rotational movement. Concerning systems employing larger angular steps the variance reluctance stepper motor is profitable and for applications requiring smaller steps the hybrid stepper motor is advantageous [3]. The corresponding static torque curve to Figure 2.3, used to describe the behaviour of the stepper motor, is shown in Figure 2.4, where the individual static torque of each phase is illustrated together with the sum of the phase torque components. The curve is obtained by altering the rotor position with mechanical angle θ m and measure the torque generated by each phase at the given angle [10]. Both current direction and timing in the winding determine the performance of the stepper motor and the total system, making the maximum torque limited with less winding current [2]. Respective phase torque components shown in the figure are 5

16 2. Valve Control System described as τ A = τ ka cos(θ m ) = ki A cos(n 1 θ m ), τ B = τ kb sin(θ m ) = ki B sin(n 1 θ m ), (2.2) where I A = I B = constant, k a scaling factor, θ m is the mechanical rotor position at 0 in Figure 2.4, and N 1 is a constant depending on the motor design. For the example in Figure 2.4 N 1 is 3, for one mechanical turn the rotor position will be repeated three times. Each torque component contributes to the total torque as τ tot = τ A + τ B. (2.3) Assuming a full alignment of phase B at 0 and consequently a 90 misalignment of phase A in the same position, phase B and A are positioned in equilibrium at 0 respectively 30 as shown in Figure 2.4. Equilibrium defines the position where the rotor and stator teeth are aligned [2], i.e. when the energised winding is aligned with the opposite polarity of the rotor as shown in Figure 2.3. At angle θ k the highest torque component τ k is generated. The component τ k, also known as the motor pull out torque, is dependent on the motor speed [10]. Using a τ k -speed curve the relation between maximum torque compared to the applied load torque at a given speed is expressed. Additionally presented in a τ k -speed curve are the motor resonance and instability problems occurring at specific velocities [2]. Consequently, using a τ k - speed curve additional motor behaviour and dependencies can be observed. 1 H ( k, k ) A B A+B Static Torque [Nm] Mechanical Angle [deg] Figure 2.4: Static torque curve of 2-phase motor. As torque is a result of the motor winding energisation the sequence and amplitude of the current fed to the windings are essential. The easiest stepping sequence 6

17 2. Valve Control System is single stepping where the rotor move in a number of distinct position during one electrical period [2]. The static torque curves shown in Figure 2.4 describe the torque components of the motor structure provided in Figure 2.3, whereas the position illustrated in Figure 2.3 refers to the static torque curves at 0 mechanical degrees. In that position winding B is fed a positive current, indicating phase B+ having a north polarity. At position 30, a positive current is instead fed to winding A, moving the rotor another step counterclockwise. The two remaining movements of the electrical turn are performed by in turn energising winding B with a negative current, followed by an applied negative current in winding A at 60 respectively 90. The described sequence is denoted as B+, A+, B, A [10]. For every stepwise rotation of the rotor, the teeth will oscillate around the new position before settling. These oscillations can be viewed as a damped oscillating system and every new step will result in the same damped oscillations [10]. The step length is described by motor geometry according to step length = 360 N r N s, (2.4) where N r is the number of rotor teeth and N s implies the number of phases in the stator [2]. The two states and six rotor teeth shown in Figure 2.3 result in a step length of 30. Increasing the step resolution results in a smoother shaft rotation. After single stepping half stepping is the next level to an increased step resolution. Half stepping is introduced by repeating having one phase on and afterwards two phases on simultaneously [2]. To further increase step resolution the excitation mode microstepping is applied. As stepper motors with a high number of stator and rotor teeth are complicated to manufacture it is advantageous to divide the rotating steps into subsets. Such an action is achieved by applying winding currents unequal in amplitude while energising two phases simultaneously. This technique requires a closer surveillance as feedback to control varying motor parameters such as temperature [2]. With a position dependent current expressed as I A = Îsin(θ e), I B = Îcos(θ e), (2.5) the current together with (2.3) is altered as τ tot = ki A sin(θ m ) + ki B cos(θ m ) = kîcos(θ m θ e ), (2.6) creating a motor torque dependent on both electrical position, θ e, and mechanical rotor position, θ m, resulting in an increased stepping resolution. Adding a load torque result in an angular displacement of the equilibrium position and misalignment of the rotor and stator, a disadvantage associated with the stepper motor [2]. This step position error is inevitable, but by designing a steep curve around the step position the effects are decreased. Choosing a motor with higher τ k than necessary is another option to increase the steepness of the curve, separating the actual curve from the sinusoidal approximation [2]. 7

18 2. Valve Control System The holding torque τ H is the maximum achievable torque produced by the motor before it starts an uncontrolled continuous rotation. This parameter is important as it provides an overall estimation of the system capability [2]. As shown in Figure 2.4 the total torque produced by the motor at a given rotor angle is a sum of the individual phase contributions. For a two phase motor τ H is derived from τ ka and τ kb as [10] τ H = 1 τ ka + 1 τ kb = 2τ k, (2.7) 2 2 hence τ k is defined as the effective value of τ H Hybrid Stepper Motor Since a variance reluctance stepper motor require at least three phases while the investigated valve control system is designed to use a two phase stepper motor, a hybrid stepper motor is consequently studied in this project [5]. The hybrid stepper has the combined ability of both permanent magnet and variable reluctance stepper motor. The stator is divided into poles, each with a surrounding coil [10]. As the rotor of the hybrid stepper is a cylindrical permanent magnet with opposite polarised edges it react to the produced magnetic field which results in a rotational movement. An overview of the rotor is shown in Figure 2.2, where the structure and polarisation distribution is defined. In order to provide a controllable stepwise rotation of the rotor, the rotor and stator poles are divided into small sections i.e. teeth, which are noticeable in Figure 2.2. Shown in the figure, the misalignment of the north and south rotor teeth is 90. A motor with four stages of phase excitation as shown in Figure 2.4 the relation of step length described in (2.4) is reduced to [3] step length = 90 N r, (2.8) an equation true for all 2-phase bipolar hybrid stepper motors. With N r = 50, one mechanical turn of a 2-phase stepper motor is the result of 50 teeth alignment periods. With one teeth alignment period being equal to one electrical period θ e = 360 = 7.2 making one electrical period consisting of 4 steps making one step 50 length 1.8. The mechanical angle in relation to the electrical angle is thereby derived as θ e = 50 θ m, indicating that one mechanical turn is translated into 50 electrical periods. For a hybrid stepper motor the typical step length is 1.8 which provide a higher angular position resolution compared to a variable reluctance motor [3] Electrical Model Of The Hybrid Stepper Motor The electrical model of the hybrid stepper motor refer to the winding circuit illustrated in Figure 2.5. The figure shows a simplified schematic of a basic winding circuitry where the electric dynamics are mathematically described as 8 v j = Ri j + e j, (2.9)

19 2. Valve Control System with R being the winding resistance, v j the applied phase voltage, i the system current and e j the EMF for a specific phase j. The induced voltage depend on the flux linkage as e j = dψ j dt, (2.10) consisting of both the winding inductance voltage and the induced back-emf, e bj, seen in Figure 2.5. i j R L + + ω m θ m v j e j e bj + Figure 2.5: Circuitry of the stator windings, where resistance, inductance, EMF and back-emf are present, in addition to the winding applied input voltage and the resulting winding current. For a 2-phase stepper motor the total flux linkage in phase A and B are described as Ψ A = L A (θ e, i)i A + Ψ M cos(θ e ), (2.11) Ψ B = L B (θ e, i)i B + Ψ M sin(θ e ) where L A and L B are the position and current dependent inductance in winding A respectively B. The currents i A and i B flows in respective phase while Ψ M is the maximum flux linkage from the permanent magnet through the winding. The angle of the electrical period is denoted θ e. By combining (2.10) and (2.11) the phase A electrical dynamics are defined as v A = Ri A + dψ A dt = Ri A + d dt (L A(θ e, i)i A + Ψ M cos(θ e )) = Ri A + L A (θ e, i) di A dt + i dl A (θ e, i) A + d dt dt (Ψ Mcos(θ e )) = Ri A + L A (θ e, i) di A ( dt LA (θ e, i) θ e + i A θ e t + L ) A(θ e, i) i Ψ M ω e sin(θ e ). i t (2.12) The electrical dynamics for phase B is determined in the same manner and expressed as v B = Ri B + L B (θ e, i) di ( B dt + i LB (θ e, i) θ e B θ e t + L ) B(θ e, i) i + Ψ M ω e cos(θ e ). i t (2.13) 9

20 2. Valve Control System As a result of the permanent magnet in the hybrid stepper rotor there exists a magnetic force unaffected of the winding energisation called detent torque τ d [11]. Shown in Figure 2.6, τ d exists between the stator and rotor adjacent teeth where the latter attracts the former due to reluctance. With one electrical period being 7.2 of one mechanical period, as described in Section 2.1.1, together with a step length of 1.8, there exist four steps within a period. One electrical period start with rotor and stator teeth being aligned as shown in Figure 2.6 and ends when the rotor teeth has moved four quarters. Figure 2.6: Full alignment of stator and rotor teeth. One electrical turn will therefor result in four positions where the detent torque is maximal as shown in Figure 2.7. An assumption of a periodical detent torque is due to the expectation of the teeth being equally separated. The detent torque is defined as [12] τ d (θ e ) = A d sin(4θ e ), (2.14) with A d being the amplitude of the position dependent detent torque Mechanical Model Of The Motor And Load The mechanical model of a motor is derived from its mechanical dynamics. Concerning a hybrid stepper motor the mechanical dynamic structure is set as dω m = 1 dt J (τ e τ d (θ m ) τ l ), dθ m = ω m, dt (2.15) with J being the motor inertia, τ e, τ d (θ m ) and ω m the electrical torque, detent torque depending on the mechanical rotor angle and the mechanical speed respectively [12]. A mathematical development of τ e is provided in Chapter 3. Torque component τ l is the load torque of the system, generated by the valve, the gear box and other mechanics in between [12]. 10

21 2. Valve Control System Normalised Total Voltage Ideal Normalised Detent Torque Electric Angle [deg] Figure 2.7: Wave form of the normalised detent torque and normalised total phase voltage over one electrical period. 2.2 Controller and Drive Circuit The stepper motor is digitally regulated by a controller unit sending pulses to the motor driver circuit [5]. Due to digital logic control problems appearing at moments such as start, stop, acceleration and deceleration, the design of the driver circuit is important when developing a stepper motor system. The controller can be applied as an application specific integrated circuit (ASIC), on a microprocessor or other logic integrated circuit (IC) [10]. The advantage of a microprocessor is the reprogrammability, a quality that makes it slower in comparison to an ASIC, as this logic construction is optimised for the given application. In the elaborated system the drive circuit act as a bridge between the controller and the actual motor. Within the driver circuit several modules are integrated, which are all resulting in a control of the winding current. Depending on the stepper motor, two different driver types are used. The drives are divided as bipolar and unipolar, and differs in the number of used windings and energisation sequence. The unipolar drive use four windings, each attached to a positive power supply which in turn is connected to a centre tap. Each winding is connected to ground by a switch, controlling the current flow from the centre tap through the windings in a given order making each winding being energised in turn. An example of the sequence of energising is illustrated in a timing diagram shown in Figure 2.8 where A, B, C and D are the separate windings i.e. the separate phases. The unipolar control ensures the rotor to move one position at a time and is advantageous in applications with a variance reluctance stepper motor due to the direction of the current not being relevant when creating the torque [3, 5]. 11

22 2. Valve Control System Figure 2.8: Timing diagram of a unipolar motor control sequence. A bipolar two phase control is utilised by turning on and off switches in two H- bridges controlling the direction of the current flow in the motor windings. Figure 2.9 illustrates the A phase drive configuration where switch S1 and S4 as well as switch S2 and S3 work in pairs. When S1 and S4 are turned on the voltage and current through the winding inductance will be positive. With S2 and S3 turned on the voltage and current flow in the opposite direction. The switch pairs cannot be turned on at the same time, but by being off simultaneously the output current will reach zero. This is necessary as the inductance current can not change direction instantaneously and the stored energy in the inductance is dissipated through the diodes during off-time. Current is controlled by altering the voltage on the windings though Pulse Width Modulation, PWM. Figure 2.9: A phase generated from H-bridge in the driver circuit. The stepwise energisation of the two windings establish a timing diagram shown in Figure 2.10 where a high value indicates a positive current in the winding and a low value represents a negative phase current with full stepping driver 12

23 2. Valve Control System mode[10]. Due to the direction of the current the windings will be energised in a specific manner providing a four step rotation of the rotor. The bipolar drive is suitable for the hybrid stepper motor [5]. Figure 2.10: Timing diagram of a bipolar motor control sequence. 2.3 The Valve And Gearbox The load of the valve control control system consist of a gearbox and valve and the torque from both components contributes to the load torque τ l. The valve in this valve control system is a butterfly valve which open and closes by 90. It has a compact design requiring a small installation space making it advantageous in space limited applications [13]. The gearbox is connected between the motor shaft and the load. It converts the motor torque and speed output to the high torque and low speed required by the valve, enabling the possibility of using smaller motor. Depending on the torque τ e produced by the motor and the torque required by the valve τ v the ratio between the two torques, the gear ratio n G, is obtained as τ v = n G τ e (2.16) assuming a positive and constant gear rate [14]. The valve is used to control liquid flow. Its main components are the body, stem, and disc mounted as shown in Figure 2.11, provided by Johan Deger at LK Valves. Leakages are minimised as the disc closing tightly around the lining. Two components creates the total valve torque as τ v = τ c + τ a, (2.17) where τ c is the torque originating from construction and τ a is the aerodynamic torque coefficient component. Construction torque τ c depends on construction parameters such as friction and will increase with age [15, 16]. This is why having a motor able to produce more torque than nominally necessary is important, to count for increased construction torque and thereby extend the life time of the motor and valve system. The fluid flowing in the pipe applies pressure on the disk causing the second torque component. This additional aerodynamic valve torque depend on the disk angle and geometry of the nearby pipe. It is described by a normalised aerodynamic torque coefficient component τ a. When the disk is closed and positioned at 0, τ a is generally small due to equal forces on both sides of the disk. Opening the disk increase the contribution from the fluid, changing τ a. At 80 the aerodynamic valve torque reach its maximum value, corresponding to the position where the fluids in the pipe has the largest impact on the disk. In the end of the opening process, τ a decrease to arrive at a small and balanced value at a fully opened disk at 90 [16, 17]. 13

24 2. Valve Control System Figure 2.11: CAD model of the butterfly valve provided by Johan Deger at LK Valves. 14

25 3 Load Torque Estimation Torque is the rotational analogue to force, measured in newton meter and translating Newtons second law F = ma into τ = J θ = J dω dt, (3.1) where J is the body inertia equivalent the mass m and θ the angular acceleration equivalent the linear acceleration a. Superposition is utilised to measure a net torque, hence the total torque of a motor is described as (2.15), relating the produced electrical torque τ e to the detent torque τ d (θ e ) and the load torque τ l. Rewriting (2.15) results in τ l = τ e τ d (θ e ) J dω m dt Bω m, (3.2) which additionally includes the viscous damping constant B [9]. This constant denotes for the friction and second order effects due to eddy currents and hysteresis in the motor [9]. Noticeably, the produced electromagnetic torque is equal to the sum of the applied load torque and the detent torque when the rotor speed is zero. With an assumption of a constant rotational speed, i.e. the stepping of the motor is smooth enough to be seen as uninterrupted reduce (3.2) to τ l = τ e τ d (θ e ) Bω m. (3.3) Estimating τ l by evaluating the detent torque, loss component Bω m and motor torque for the specific system enables the possibility to monitoring the valve health without the usage of a torque sensor. Both B and the detent torque are small components compared to the motor torque, why the latter is the main focus when trying to estimate τ l Phase Stepper Motor Torque Expression An expression regarding the torque produced by a 2-phase hybrid stepper motor is initiated using the rule of energy conservation in the coupling field. The same reasoning is applicable to a motor with more phases. For the 2-phased motor the differential form of the comprise dw e = dw f + dw m, (3.4) 15

26 3. Load Torque Estimation where dw e is the electrical energy applied to the coupling field, dw f the energy from electrical system reserved in the coupling field and the mechanical energy dw m moving the rotor [5]. Dividing the electrical input energy dw e in its fundamental components, it can be expressed as dw e = (e A i A + e B i B )dt, (3.5) where e A and e B are the induced EMF from a change in flux concerning each phase as shown in (2.10) and i A and i B are the respective phase currents [5]. The relation between mechanical energy and torque τ e is described as as [5] dw m = τ e dθ m. (3.6) Fixing the rotor in a set position making dθ m = 0 and causing dw m to be zero, entails (3.4) and (3.5) to merge in [5] dw f = dw e = (e A i A + e B i B )dt. (3.7) Using (2.10) in (3.7) and integrating over the rotor angles, the total amount of energy stored in the coupling field is described as W f = ΨA 0 i A dψ A + ΨB 0 i B dψ B. (3.8) With the rotor rotating a small distance dθ e together with an assumption of constant currents during this period, (3.4), (3.6) and (3.8) form i A dψ A + i B dψ B = dw e = dw f + τ e dθ m. (3.9) With dθ m approaching zero, (3.9) is reformulated, expressing the motor torque as τ e = W f θ m + i A Ψ A θ m + i B Ψ B θ m = W f θ m, (3.10) where the latter term W f is the coenergy of the system. Coenergy is a concept not having a physical aspect, however being used to express electromagnetic force [18]. In a motor assumed not to reach saturation the energy, as a function of angular position and current, is linear in the magnetisation curve [5]. Coenergy resembles the energy stored in the coupled field and is expressed as W f = ia 0 Ψ A di A + ib The flux linkage will then also be linear to the current as 0 Ψ B di B. (3.11) Ψ A = L A (θ e )i A + Ψ M cos(θ e ) Ψ B = L B (θ e )i B + Ψ M sin(θ e ), (3.12) where L A (θ e ) respectively L B (θ e ) are the inductance of each phase depending on electrical angle θ e [5]. Inserting (3.12) in (3.11) together with θ e = N r θ m, then 16

27 3. Load Torque Estimation (3.10) results in τ e = 1 L A (N r θ m ) 2 i2 A θ m + 1 L B (N r θ m ) 2 i2 B θ m } {{ } Reluctance Torque N r Ψ M i A sin(n r θ m ) + N r Ψ M i A cos(n r θ m ), }{{} Permanent Magnet Torque (3.13) where the former segment of the equation is the contribution from the reluctance torque and the latter is the contribution from the permanent magnet. As shown in (3.13) the contribution from the reluctance torque is depending on a rotor position dependent inductance. With aligned stator and rotor teeth the reluctance is low introducing a high flux at a given current, enabling a high inductance. At a misaligned position, the reluctance at a given current is higher resulting in a lower inductance. Due to the construction of the stator and rotor teeth, the inductance is approximately periodic with an average phase inductance L 0 and a rotor position dependent inductance variation L 1 [3]. Concerning the 2- phase stepper motor the A and B position dependent inductance are 180 misaligned from each other and expressed as L A (θ m ) = L 0,A + L 1,A sin(2n r θ m ) L B (θ m ) = L 0,B L 1,B sin(2n r θ m ). (3.14) Using (3.14) in (3.13) the torque expression for a 2-phase stepper motor is derived as τ e = ( i 2 AN r L 1,A cos(2n r θ m ) i 2 BN r L 1,B cos(2n r θ m ) ) +N r Ψ M ( i A sin(n r θ m ) + i B cos(n r θ m )). 3.2 Motor Torque Estimation Using Power (3.15) Another estimate of τ e can be derived from output power and rotational speed as τ e = P out ω m. (3.16) In a realistic model of the motor the system losses need to be accounted for, as they are present during operation. Thereby, the input power can be related to the output power as P out = P in P loss. (3.17) Calculating the output power for a 2-phase stepper motor is executed using (2.12) and (2.13). Ignoring the saturation effect, leaves only a position variance [8]. The input power to a system is then used to extract the electrical torque of the stepper motor. Concerning phase A such an operation results in P in,a = v A i A = Ri 2 A + L A (θ e ) di A dt i A + ω e ( i 2 A ) L A (θ e ) i A Ψ M sin(θ e ), θ e (3.18) 17

28 3. Load Torque Estimation while phase B resulting in P in,b = v B i B = Ri 2 B + L B (θ e ) di B dt i B + ω e ( i 2 B ) L B (θ e ) + i B Ψ M cos(θ e ). θ e (3.19) Combined, (3.18) and (3.19) is described as P in,2-phase = P ina + P in,b = R(i 2 A + i 2 B) + L A (θ e ) di A dt i A + L B (θ e ) di B + ω m N r (i 2 L A (θ e ) A i A Ψ M sin(θ e ) + i 2 B θ e dt i B L B (θ e ) θ e + i B Ψ M cos(θ e ) (3.20) with the the last term being P out according to (3.17) using term identification. Identifying the motor torque component from (3.13) and rewriting (3.20) as (3.16) generates P in,2-phase = P ina + P in,b ), = R(i 2 A + i 2 B) + L A (θ e ) di A dt i A + L B (θ e ) di B dt i B ) + ω m (τ e i2 A L A (θ e ) θ e i2 B L B (θ e ) θ e To describe the generated motor torque in terms of input power and losses, (3.21) result in τ e = 1 ω m ( P in,2-phase R(i 2 A + i 2 B) L A (θ e ) di A dt i A L B (θ e ) di B ( 1 L A (θ e ) 2 i2 A + 1 θ e 2 i2 B ) L B (θ e ), θ e. dt i B ) (3.21) (3.22) as an additional motor torque expression in comparison to (3.13), having both position and rotational speed dependencies. Due to (3.14), (3.23) the torque expression for a 2-phase stepper motor is written as τ e = 1 ( P in,2-phase R(i 2 A + i 2 ω B) L A (θ e ) di A m dt i A L B (θ e ) di ) B dt i B ( N r i 2 AL 1,A cos(2n r θ m ) N r i 2 BL 1,B cos(2n r θ m ) ). (3.23) 18

29 4 Motor and Measurement Setup In this section the employed stepper motor is described together with measurement setups to determine different motor specific parameters. 4.1 Stepper Motor System In Thesis The stepper motor under investigation is module PD from Trinamic, comprising a controller, driver and motor. Specifications for the module are collected from datasheets with an summary of important parameters stated in Table 4.1 and a simplification of the module implementation with specific processors and motor shown in Figure 4.1 [19, 20, 21]. The implementation consists of the 2-phase bipolar hybrid stepper motor QSH from Trinamic together with a controller unit, driver stage and the sensostep magnetic encoder to detect step [22]. The controller unit is implemented with microcontroller ARM Cortex-M3, motion controller circuit TMC429, pre-driver circuit TMC262 and a TMCL-memory [21]. Dividing the controller design in subdivisions establish a possibility for the microcontroller to control three different stepper motors simultaneously [23]. A MOSFET drive circuit implemented with H-bridges transfer the required currents to the motor windings. Typically the applied phase voltage has a value of 12, 24 or 48 V although the motor can withstand a maximum value of 51 V [20]. With the number of rotor teeth being 50, each full step length is equal to 7.2, corresponding to 1 of one mechanical turn. 50 The motor controller unit enables each full step to be divided into 256 microsteps. It also has an implemented current control, spreadcycle, which behavioural characteristics are based on the chopper principle [22]. The chopper regulate the current to match a reference current by adjusting the applied phase voltage, determining the current slope. Trinamics own coolstep function is additionally implemented, adapting the winding current level depending on the load to minimise the power consumption and place the module in energy saving mode when steady state operation is reached [22]. By controlling the power transmission the efficiency is maximised and heating problem decreases. Benefits arising from higher efficiency is more perceptible with an increased rotational speed. As the coolstep function require knowledge about the applied load torque the feature stallguard2 is integrated in the controller unit [22]. This function measure the back-emf level and translate the result to a corresponding load torque value for the coolstep function. The feature is used to locate the load torque value both for the coolstep function and to monitor the load torque and thereby prevent motor failure due to overload. Another feature implemented a microplayer which increase the smoothness in ro- 19

30 4. Motor and Measurement Setup Figure 4.1: Simplified circuit description of motor module PD with specifications stated in [21] and [20]. tation and enable microstepping [22]. Motor systems from Trinamic are operated using the combined protocol and programming language Trinamic Motion Control Language TMCL. TMCL for motor applications is developed using a PC-based integrated development environment from Trinamic TMCL-IDE, connected to the motor through USB or field buses CAN, RS232 or RS485 [21]. Commands can be stored in the application and performed as stand-alone operation or be operated directly from a host. The TMCL-IDE furthermore offers a possibility to graphically monitor motor behaviour and parameters in real time. A large range of parameters can be altered to achieve an optimal application setting. However, in the project the standard values are considered and no fundamental characteristics have been changed. The 2-phase bipolar hybrid stepper motor controls a butterfly valve from LK Valves with a holding torque of 250 Nm. According to Johan Deger at LK Valves, a gearbox with gear ratio 84:1 is used in the system to accomplish this torque. The gear box is supposed to be ideal, why the load torque of the system τ l equals the torque acquired by the valve τ v. Furthermore the gearbox should transfer a rotor speed of 21 rpm 97 rpm to a valve speed of 0.3 rpm 1.2 rpm with an actuation time of 13 60s. Due to the torque of the valve increasing with age, the motor should be able to deliver a torque to move a load requiring 300 Nm, having a safety margin of 20%. 20

31 4. Motor and Measurement Setup Table 4.1: Specifications for Trinamic motor module PD [19, 20]. Dimension Nema24 #Rotor Teeth 50 Rated Current, i rms 2.8 A Peak Current, i maxpeak 4 A Phase Voltage, v max 51 V Phase Voltage, v typ 12, 24, 48 V Holding Torque, τ H 3.1 Nm Phase Resistance at 20, R 1.5Ω Phase Inductance (typ.), L 6.8 mh Rotor Inertia, J 840 g cm Motor Parameter Measurement Setup Some parameters required for the theoretical model and torque estimators were not provided by the manufacturer and datasheet, whereas actual measurements on the real motor were necessary. Experiments and measurements were also conducted to verify the information and graphs provided by the datasheet Winding Resistance In order to obtain an accurate model of the motor the winding resistance, seen in Figure 2.5, needs to be determined. This was done using a multimeter. The resistance measurement required the windings to be open circuited, as the measurement probes were connected according to Figure 4.2. i i R L Multimeter e bi + Figure 4.2: Measurement setup for winding resistance determination. The multimeter was connected to the open circuit winding. 21

32 4. Motor and Measurement Setup Maximum Flux Linkage To estimate the electrical torque given as (3.13) one of the motor specific parameters to determine is the maximum flux linkage Ψ M. As e = dψ, knowing the flux dt linkage from each phase from (3.12), the expressions for each phase induced voltage contribution are described as e A = dψ A dt e B = dψ B dt = L A (θ e ) di A dt + i L(θ e ) A = L B (θ e ) di B dt + i B θ e L(θ e ) θ e θ e t Ψ Mω m sin(θ e ), θ e t + Ψ Mω m cos(θ e ). (4.1) Open circuit measurements result in zero current through the windings, establishing the induced voltage being equal to the input voltage, arriving in e A = dψ A dt e B = dψ B dt = Ψ M ω m sin(θ e ), (4.2) = Ψ M ω m cos(θ e ). (4.3) The induced voltage was created by rotating the motor shaft in a known and constant speed. By extracting the voltage curves over time using a digital oscilloscope, the mechanical speed was derived. Knowing the induced voltage over time the exact speed of the shaft rotation was established using the assumption of the induced voltage being two phase and offset by 90. Such an estimation creates the opportunity of constructing the two phases as a complex voltage vector V = V A + jv B. The absolute of the created complex vector is the length of the vector and the amplitude of the induced voltage. The mean value of the vector during a set number of times will create a linear model whose slope is the frequency of the system. From this frequency the correct speed of the rotation was calculated and the value of Ψ M determined Position Dependent Inductance The rotor position dependent inductance was estimated applying the configurations shown in Figure 4.3. When conducting the experiment an external arm was attached perpendicular to the rotor shaft. As shown Figure 4.3 the arm had an equal length x on both sides of the shaft and an evenly distributed weight to minimise the impact on the rotor. Manual adjustment of the rotor angle was obtained by altering the position by y length units. With a measuring stick at a given length from the shaft, the rotational angle θ of the motor was with these lengths calculated using trigonometric identities. The inductance values were extracted by connecting a LCR meter to the windings and measure the inductance at distinct rotor angles. The used equipment was an Atlas LCR model LCR40, with an accuracy of ±1.5% concerning inductance measurements [24]. In order to obtain conclusive results the experiment was done for both sides of the rotor, i.e. when the rotational movement was both clockwise and counter clockwise. 22

33 4. Motor and Measurement Setup Figure 4.3: Setup of the position dependent inductance measurement. 4.3 Detent Torque Measurement Setup In order to measure the detent torque the rotor was rotated by an external force. The motor was therefor disconnected from its digital control to ensure the measurements occurred without impact from other sources of force. To create a controlled rotation of the rotor an arm was placed perpendicular to the shaft. The procedure to extract the rotor angle θ m were the same as to the measurements of the position dependent inductance in Section At a given angular position the weight this rotational movement caused on the arm on the opposite side of the shaft was measured. The setup of the detent torque measurement is shown in Figure 4.4. Figure 4.4: Setup for detent torque measurement. Using the length l from the rotor centre to the point where the weight m was measured the torque on the shaft at a specific angular position is calculated as τ = m g l with g being the gravitational acceleration of 9.81 m/s 2. The sampling 23

34 4. Motor and Measurement Setup frequency of the data, i.e. the data points analysed during the experiment, needs to fulfil the requirement of the specific motor being bipolar and having 2 phases together with one electrical turn being 7.2. This translates into that for one electric period there must be at least two measurement points. 24

35 5 Extracted Stepper Motor Parameters The following chapter contain results of performed motor measurements required to obtain a correct model of the system. 5.1 Motor Torque Measurements Winding Resistance The individual resistance of each winding was measured using a multimeter as illustrated in Figure 4.2. As a result, measurements concluded that both windings had a resistance value of 1.4 Ω Maximum Flux Linkage Exposing the shaft of the investigated motor to a mechanical speed cause an induced voltage of each phase as shown in Figure 5.1. As assumed in Section the two voltages are offset by 90 why the introduction of the complex voltage vector is accurate. Calculation of the complex voltage angle from such an estimation creates the model shown in Figure 5.2. Linearising this model results in the linear equation of y = 1735x From the slope of this straight line the mechanical speed of the system is determined as rpm, a value close to the, by a tachometer, measured rpm. Rearranging (4.1) to describe a cosine function with amplitude Ψ m the flux linkages is additionally plotted in Figure 5.1. The maximum flux linkage Ψ m is determined to be Wb Position Dependent Inductance Measurements of the position dependent inductance resulted in the graph shown in Figure 5.3. Although some measured values are misplaced, the winding inductance can be mathematically described as a sine. Due to one electrical period being 7.2 and the bipolar drive of the investigated motor, two periods of the inductance variation are seen in the figure. The 180 shift of the measured phase inductance is explained by the 90 phase misalignment and the inductance position dependency provided in (3.14). Presented in Table 5.1 are the key values extracted from the measurements such as the maximum, minimum and mean inductance values in addition to the 25